空气动力学英文课件Chapter_02

nullPART IPART IFUNDAMENTAL PRINCIPLES （基本原理）In part I, we cover some of the basic principles that apply to aerodynamics in general. These are the pillars on which all of aerodynamics is basedChapter 2Chapter 2Aerodynamics: Some Fundamental Principles and EquationsThere is so great a difference between a fluid and a collection of solid particles that the laws of pressure and of equilibrium of fluids are very different from the laws of the pressure and equilibrium of solids . Jean Le Rond d’Alembert, 17682.1 Introduction and Road Map2.1 Introduction and Road MapPreparation of tools for the analysis of aerodynamics Every aerodynamic tool we developed in this and subsequent chapters is important for the analysis and understanding of practical problems Orientation offered by the road map 2.2 Review of Vector relations2.2 Review of Vector relations2.2.1 to 2.2.10 Skipped over 2.2.11 Relations between line, surface, and volume integralsThe line integral of A over C is related to the surface integral of A(curl of A) over S by Stokes’ theorem:Where aera S is bounded by the closed curve C:nullThe surface integral of A over S is related to the volume integral of A(divergence of A) over V by divergence’ theorem:Where volume V is bounded by the closed surface S:If p represents a scalar field, a vector relationship analogous to divergence theorem is given by gradient theorem:2.3 Models of the fluid: control volumes and fluid particles2.3 Models of the fluid: control volumes and fluid particlesImportance to create physical feeling from physical observation. How to make reasonable judgments on difficult problems. In this chapter, basic equations of aerodynamics will be derived. Philosophical procedure involved with the development of these equationsnullInvoke three fundamental physical principles which are deeply entrenched in our macroscopic observations of nature, namely, a. Mass is conserved, that’s to say, mass can be neither created nor destroyed. b. Newton’s second law: force=mass☓acceleration c. Energy is conserved; it can only change from one form to another 2. Determine a suitable model of the fluid. 3. Apply the fundamental physical principles listed in item 1 to the model of the fluid determined in item2 in order to obtain mathematical equations which properly describe the physics of the flow. nullEmphasis of this section: What is a suitable model of the fluid? How do we visualize this squishy substance in order to apply the three fundamental principles? Three different models mostly used to deal with aerodynamics. finite control volume （有限控制体） infinitesimal fluid element （无限小流体微团） molecular （自由分子） null2.3.1 Finite control volume approachDefinition of finite control volume: a closed volume sculptured within a finite region of the flow. The volume is called control volume V, and the curved surface which envelops this region is defined as control surface S. Fixed control volume and moving control volume. Focus of our investigation for fluid flow.nullnull2.3.2 Infinitesimal fluid element approachDefinition of infinitesimal fluid element: an infinitesimally small fluid element in the flow, with a differential volume. It contains huge large amount of molecules Fixed and moving infinitesimal fluid element. Focus of our investigation for fluid flow.nullThe fluid element may be fixed in space with fluid moving through it, or it may be moving along a streamline with velocity V equal to the flow velocity at each point as well.null2.3.3 Molecule approachDefinition of molecule approach: The fluid properties are defined with the use of suitable statistical averaging in the microscope wherein the fundamental laws of nature are applied directly to atoms and molecules.In summary, although many variations on the theme can be found in different texts for the derivation of the general equations of the fluid flow, the flow model can be usually be categorized under one of the approach described above.null2.3.4 Physical meaning of the divergence of velocityDefinition of : is physically the time rate of change of the volume of a moving fluid element of fixed mass per unit volume of that element.Analysis of the above definition: Step 1. Select a suitable model to give a frame under which the flow field is being described. a moving control volume is selected.nullStep 2. Select a suitable model to give a frame under which the flow field is being described. a moving control volume is selected.Step 3. How about the characteristics for this moving control volume? volume, control surface and density will be changing as it moves to different region of the flow.Step 4. Chang in volume due to the movement of an infinitesimal element of the surface dS over . nullnullThe total change in volume of the whole control volume over the time increment is obviously given as bellowStep 5. If the integral above is divided by .the result is physically the time rate change of the control volume nullStep 6. Applying Gauss theorem, we have Step 7. As the moving control volume approaches to a infinitesimal volume, . Then the above equation can be rewritten as nullAssume that is small enough such that is the same through out . Then, the integral can be approximated as , we have orDefinition of : is physically the time rate of change of the volume of a moving fluid element of fixed mass per unit volume of that element.nullAnother description of and :Assume is a control surface corresponding to control volume , which is selected in the space at time . At time the fluid particles enclosed by at time will have moved to the region enclosed by the surface . The volume of the group of particles with fixed identity enclosed by at time is the sum of the volume in region A and B. And at time , this volume will be the sum of the volume in region B and C. As time interval approaches to zero, coincides with . If is considered as a fixed control volume, then, the region in A can be imagined as the volume enter into the control surface, C leave out.nullnullBased on the argument above, the integral of can be expressed as volume flux through fixed control surface. Further, can be expressed as the rate at which fluid volume is leaving a point per unit volume.nullThe average value of the velocity component on the right-hand x face isThe rate of volume flow out of the right-hand x face isThat into the left-hand x face isThe net outflow from the x faces is per unit time nullThe net outflow from all the faces in x,y,z directions per unit time is The flux of volume from a point is null2.4 Continuity equationIn this section, we will apply fundamental physical principles to the fluid model. More attention should be given for the way we are progressing in the derivation of basic flow equations. Derivation of continuity equation Step 1. Selection of fluid model. A fixed finite control volume is employed as the frame for the analysis of the flow. Herein, the control surface and control volume is fixed in space.nullnullStep 2. Introduction of the concept of mass flow. Let a given area A is arbitrarily oriented in a flow, the figure given bellow is an edge view. If A is small enough, then the velocity V over the area is uniform across A. The volume across the area A in time interval dt can be given asnullThe mass inside the shaded volume isThe mass flow through is defined as the mass crossing A per unit second, and denoted as or nullThe equation above states that mass flow through A is given by the product Area X density X component of flow velocity normal to the areamass flux is defined as the mass flow per unit areanullStep 3. Physical principle Mass can be neither created nor destroyed.Step 4. Description of the flow field, control volume and control surface.Directional elementary surface area on the control surfaceElementary volume inside the finite control volumenullStep 5. Apply the mass conservation law to this control volume.Net mass flow out of control volume through surface STime rate decrease of mass inside control volume VorStep 6. Mathematical expression of BThe elemental mass flow across the area is The physical meaning of positive and negative of nullnullThe net mass flow out of the whole control surface S Step 7. Mathematical expression of CThe mass contained inside the elemental volume V is The mass inside the entire control volume is nullThe time rate of increase of the mass inside V is The time rate of decrease of the mass inside V is Step 8. Final result of the derivation Let B=C , then we get nullorDerivation with moving control volumeMass at time Mass at time nullBased on mass conservation lawConsider the limits as nullThen we get the mathematical description of the mass conservation law with the use of moving control volume Why the final results derived with different fluid model are the same ??nullStep 9. Notes for the Continuity Equation above The continuity equation above is in integral form, it gives the physical behaviour over a finite region of space without detailed concerns for every distinct point. This feature gives us numerous opportunities to apply the integral form of continuity equation for practical fluid dynamic or aerodynamic problems. If we want to get the detailed performance at a given point, then, what shall we deal with the integral form above to get a proper mathematic description for mass conservation law?nullStep 10. Continuity Equation in Differential formControl volume is fixed in spaceThe integral limit is not the sameThe integral limit is the samenullorA possible case for the integral over the control volumenullIf the finite control volume is arbitrarily chosen in the space, the only way to make the equation being satisfied is that, the integrand of the equation must be zero at all points within the control volume. That is,That is the continuity equation in a partial differential form. It concerns the flow field variables at a point in the flow with respect to the mass conservation lawIt is important to keep in mind that the continuity equations in integral form and differential form are equally valid statements of the physical principles of conservation of mass.they are mathematical representations, but always remember that they speak words.nullStep 11. Limitations of the equations derivedContinuum flow or molecular flowAs the nature of the fluid is assumed as Continuum flow in the derivation soIt satisfies only for Continuum flowSteady flow or unsteady flowIt satisfies both steady and unsteady flowsviscous flow or inviscid flowIt satisfies both viscous and inviscid flowsCompressible flow or incompressiblw flowIt satisfies both Compressible and incompressiblw flowsnullDifference between steady and unsteady flow Unsteady flow:The flow-field variables are a function of both spatial location and time, that isSteady flow:The flow-field variables are a function of spatial location only, that isnullFor steady flow:For steady incompressible flow:null2.5 Momentum equationNewton’s second law whereForce exerted on a body of massMass of the bodyAccelerationnullConsider a finite moving control volume, the mass inside this control volume should be constant as the control volume moving through the flow field. So that, Newton’s second law can be rewritten asDerivation of momentum equation Step 1. Selection of fluid model. A fixed finite control volume is employed as the frame for the analysis of the flow. nullStep 2. Physical principle Force = time rate change of momentumStep 3. Expression of the left side of the equation of Newton’s second law, i.e., the force exerted on the fluid as it flows through the control volume. Two sources for this force: Body forces: over every part of V 2. Surface forces: over every elemental surface of SBody force on a elemental volume nullBody force over the control volume Surface forces over the control surface can be divided into two parts, one is due to the pressure distribution, and the other is due to the viscous distribution. Pressure force acting on the elemental surfaceNote: indication of the negative sign Complete pressure force over the entire control surfacenullThe surface force due to the viscous effect is simply expressed by Total force acting on the fluid inside the control volume as it is sweeping through the fixed control volume is given as the sum of all the forces we have analyzed Step 4. Expression of the right side of the equation of Newton’s second law, i.e., the time rate change of momentum of the fluid as it sweeps through the fixed control volume. nullMoving control volumenullLet be the momentum of the fluid within region A, B, and C. for instance,At time , the momentum inside isAt time , the momentum inside isnullThe momentum change during the time interval orAs the time interval approaches to zero, the region B will coincide with S in the space, and the two limitsnullNet momentum flow out of control volume across surface STime rate change of momentum due to unsteady fluctuations of flow properties inside VThe explanations above helps us to make a better understanding of the arguments given in the text book bellow nullNet momentum flow out of control volume across surface STime rate of change of momentum due to unsteady fluctuations of flow properties inside control volume VStep 5. Mathematical description of mass flow across the elemental area dS ismomentum flow across the elemental area dS isnullThe net flow of momentum out of the control volume through S isStep 6. Mathematical description of The momentum in the elemental volume dV isThe momentum contained at any instant inside the control volume V isnullIts time rate change due to unsteady flow fluctuation isBe aware of the difference betweenandnullStep 7. Final result of the derivation Combine the expressions of the forces acting on the fluid and the time rate change due to term and , respectively, according to Newton’s second lownullIt’s the momentum equation in integral formIt’s a vector equationAdvantages for momentum equation in integral formnullStep 8. Momentum Equation in Differential formTry to rearrange the every integrals to share the same limitgradient theoremnullcontrol volume is fixed in spaceThen we getnullSplit this vector equation as three scalar equations with Momentum equation in x direction is nulldivergence theoremAs the control volume is arbitrary chosen, then the integrand should be equal to zero at any point, that is nullx directiony directionz directionThese equations can applied for unsteady, 3D flow of any fluid, compressible or incompressible, viscous or inviscid.nullSteady and inviscid flow without body forcesnullEuler’s Equations and Navier-Stokes equations Whether the viscous effects are being considered or not Eulers Equations: inviscid flow Navier-Stokes equations: viscous flownullDeep understanding of different terms in continuity and momentum equationsnullTime rate change of mass inside control volumeTime rate change of momentum inside control volumenullNet flow of mass out of the control volume through control surface SNet flow of volume out of the control volume through control surface SNet flow of momentum out of the control volume through control surface SnullBody force through out the control volume VSurface force over the control surface SnullWhat we can foresee the applications for aerodynamic problems with basic flow equations on hand?If the steady incompressible inviscid flows are concernedPartial differential equation for velocityPartial differential equation for velocity and pressurenull2.6 An application of the momentum equation: drag of a 2D bodyHow to design a 2D wind tunnel test? How to measure the lift and drag exerted on the airfoil by the fluid? nullA selected control volume around an airfoilnullDescriptions of the control volume1. The upper and lower streamlines far above and below the body (ab and hi). 2. Lines perpendicular to the flow velocity far ahead and behind the body(ai and bh) 3. A cut that surrounds and wraps the surface of the body(cdefg)1. Pressure at ab and hi. 2. Pressure at ai and bh . ,velocity , 3. The pressure force over the surface abhinullnull4. The surface force on def by the presence of the body, this force includes the skin friction drag, and denoted as per unit span. 5. The surface forces on cd and fg cancel each other. 6. The total surface force on the entire control volume is7. The body force is negligiblenullApply to momentum equation, we havefor steady flowNote: it’s a vector equation.nullIf we only concern the x component of the equation, with represents the x component of .As boundaries of the control volume abhi are chosen far away from the body, the pressure perturbation due to the presence of the body can be neglected, that means, the pressure there equal to the freestream pressure. If the pressure distribution over abhi is constant, thennullSo thatAs ab, hi, def are streamlines, thenAs cd, fg are are adjacent to each other, thennullThe only contribution to momentum flow through the control surface come from the boundaries ai and bh. For dS=dy(1), the momentum flow through the control surface isNote: The sign in front of each integrals on the right hand side of the equation 2. The integral limits for each integrals on the right hand side of the equationnullConsider the integral form of the continuity equation for steady flow,orAs is a constant nullnullThe final result gives the drag per unit spanThe drag per unit span can be expressed in terms of the known freestream velocity and flow-field properties ,across a vertical station downstream of the body.nullPhysical meaning behind the equationMass flow out of the control volumeVelocity decrement Momentum decrement per secondnullFor incompressible flow, that is, the density is constantnull2.6.1 CommentsWith the application of momentum principle to a large, fixed control volume, an accurate result for overall quantity such as drag on a body can be predicted with knowing the detailed flow properties along the control surface. That to say, it is unnecessary to know the the details along the surface of the body.null2.7 Energy equationContinuity equation Momentum equation Unknowns: For steady incompressible invicid flows nullFor compressible flows is an additional variable, and therefore we need an additional fundamental equation to complete the system. This fundamental equation is the energy equation, which we are going to develop.Two additional flow-field variables will appear to the energy equation, that is internal energy and temperature .Energy equation is only necessary for compressible flows.nullPhysical principle(first law of thermodynamics)Energy can be neither created nor destroyed; it can only change in form Definitions of system and internal energy per unit mass eDefinition of surroundingsHeat transferred from the surroundings to the systemWork done on the surroundings by the systemnullChange of internal energy in system due to the heat transferred and the work doneAs energy is conserved, soApply the first law to the fluid flowing trough the fixed control volume, and letB1 = rate of heat added to fluid inside control volume from surroundings. B2 = rate of work done on fluid inside control volume. B3 = rate of change of energy of fluid as it flows through control volume.nullAs first law should be satisfied, thenB1+B2 = B3Actually speaking, the equation above is a power equation.Rate of volumetric heating If the flow is viscousB1 =nullRate of volumetric heating = The force includes three parts Pressure fo